document.write( "Question 846438: Make a conjecture about the following results:\r
\n" ); document.write( "\n" ); document.write( "Square root of 2 2/3=2 square root of 2/3.\r
\n" ); document.write( "\n" ); document.write( "Square root of 3 3/8=3 square root of 3/8.\r
\n" ); document.write( "\n" ); document.write( "Try to prove your conjecture. \n" ); document.write( "

Algebra.Com's Answer #513622 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!
Square root of 2 2/3=2 square root of 2/3.
\n" ); document.write( "Square root of 3 3/8=3 square root of 3/8.
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It's easy to see the pattern, except for the denominators, where\r\n" );
document.write( "I have put question marks below.  There is no way to determine just by\r\n" );
document.write( "inspection, a sequence for the denominators with just the first \r\n" );
document.write( "two terms, 3 and 8.  We could guess a general term for the\r\n" );
document.write( "denominators, but that would be risky business. \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Now let's let D be those denominators, \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and solve for D.  Square both sides:\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Multiply both sides by D\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Divide through by n\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " for n > 1\r\n" );
document.write( "\r\n" );
document.write( "So our conjecture is found by substituting that\r\n" );
document.write( "expression for D in:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " for n > 1\r\n" );
document.write( "\r\n" );
document.write( "To prove that we have to show that each side is non-\r\n" );
document.write( "negative (obvious) and have the same square.  We show \r\n" );
document.write( "that each side has the same square:\r\n" );
document.write( "\r\n" );
document.write( "LEFT SIDE = \r\n" );
document.write( "\r\n" );
document.write( "SQUARE OF LEFT SIDE = \r\n" );
document.write( "\r\n" );
document.write( "RIGHT SIDE = \r\n" );
document.write( "\r\n" );
document.write( "SQUARE OF RIGHT SIDE = \r\n" );
document.write( "\r\n" );
document.write( "So they have the same square and are both non-negative so the conjecture\r\n" );
document.write( "is proved.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );